BCSBEC crossover andsuperfluidity in atomic Fermi gases Qijin
BCS-BEC crossover andsuperfluidity in atomic Fermi gases Qijin Chen (陈启谨) Zhejiang Institute of Modern Physics and Department of Physics, Zhejiang University QC 2010 Summer School 2010 -08 -19
Acknowledgments B. Janko (Notre Dame), I. Kosztin (U Missouri – Coumbia) J. Stajic, Yan He, Chih-Chun Chien and K. Levin (U Chicago) J. E. Thomas, J. Kinast (Duke), A. Turlapov (Russia) M. Holland (JILA), M. Chiofalo (Italy), J. Milstein (U Mich) D. S. Jin, C. Regal, JT Stewart, JP Gaebler (JILA) and M. Greiner (Harvard) Cheng Chin (Chicago), Rudi Grimm (Innsbruck) Randy Hulet (Rice)
Outline n Introduction n n Theoretical formalism -- Pairing fluctuation theory n n Homogeneous case Local density approximation in a trap Population imbalance Comparing theory and experiment – Evidence for noncondensed pairs and pseudogap n n Overview of recent progress in fermionic superfluidity Tunable interactions and population imbalance in ultracold Fermi gases Overview of BCS-BEC crossover Equal spin mixture (Cuprates and Fermi gases) Momentum resolved radio-frequency spectroscopy Population imbalanced Fermi gases Phys. Rep. 412, 1 -88 (2005) Summary Science 307, 1296 (2005) Phys. Rev. Lett. 102, 190402 (2009).
Part I Introduction
Breakthroughs insuperfluidity n Discovery of superconductivity, 1911 (Onnes) n Prediction of Bose-Einstein condensation (1924) BCS theory of superconductivity, 1957 n Discovery of superfluid 3 He, 1972 n High Tc superconductors, 1986 (Bednorz and Muller) n n n BEC in dilute gases of alkali atoms (1995) Superfluidityin atomic Fermi gases (2003)
Facts About Trapped Fermi Gases n n n n Most studied: 40 K and 6 Li Confined in magnetic and optical traps Atomic number N=105 -106 Fermi temperature EF ~ 1 K Cooled down to T ~10 -100 n. K Two spin mixtures – spin “up” and “down” Tunable population imbalance Optical lattices
Bosons vs fermions E F = k. B T F T=0 spin BEC Fermi sea Pauli exclusion
Physics of BEC – Bose Statistics
Summary of BEC ………………….
Essence of Fermionic Superfluidity Increased attraction BCS-BEC Crossover fermions bosons Attractive interactions turn fermions into “composite bosons” (or Cooper pairs). These are then driven by statistics to Bose condense.
Remarkable Tuning Capability in Cold Gases via Feshbach. Resonance. Scattering length a a>0 BEC Unitary limit → ← BCS a<0 molecules > B
Crossover under control in cold Fermi atoms (1 st time possible) Magnetic Field Molecules of fermionic atoms hybridized Cooper pairs and molecules Cooper pairs k. F BEC of bound molecules Pseudogap / unitary regime BCS superconductivity Cooper pairs: correlated momentum-space pairing
BCS-BEC crossover in a nutshell
Overview of BCS theory Fermi Gas No excitation gap BCS superconductor
Zero T BCS-BEC crossover: Tuning the attractive interaction Change of character: fermionic Bosonic n (Uc – critical coupling) Unitary § Use ground state BCS-Leggett crossover wave function: Basis for : • Bd. G theory (T=0) • T=0 Gross-Pitaevskii theory in the BEC • Unequal population theories Simplicity and physical accessibility Eagles and Leggett
Thermal excitations n Pairs form without condensation pseudogap. n is natural measure ofbosonic degrees of freedom. Except in BCS Two types of excitations • Novel form of superfluidity • Never seen before, except possibly in high Tc BCS Unitary BEC
Pseudogapseen in high. TC superconductors! Pseudogap (normal state gap) is very prominent. BCS-BEC crossover physics is a possible explanation. Introducing pseudogap into Fermi gases BCS High Tc Ding et al, Nature 1996
Part II Theoretical Formalism
Introduction to BCS Theory Interacting Hamiltonian Pairing at low T BCS reduced Hamiltonian Assume Separable potential
Pair size ~ 103 – 104 º A
Bogoliubov transformation -Diagonalization Coherence factors Ground state energy
Ground state wavefunction System energy measured from bottom of band Determine the gap/order parameter BCS Gap equation
Grand canonical Hamiltonian for resonance superfluidity BCS-like ground state: Two-channel physics is not essential for BCS-BEC crossover.
Two-channel physics is not essential for wide Feshbach resonances n GB Partridge et al, PRL 95, 020404 (2005). n QC & K. Levin, PRL 95, 260406 (2005).
Grand canonical Hamiltonian BCS keeps only q=0 terms Population imbalance: Fermi gases: Take contact potential Cuprates: Separable potential:
Pairing fluctuation theory -- Physical Picture PRL 81, 4708 (1998) n n n Fermionic self energy has a pairing origin. Pairs can be either condensed or fluctuating. Condensed and noncondensed pairs do not mix. +
Equations of motion n n Factorize G 3 G and G 2 on equal footing T-matrix approximation (G 0 G scheme) Condensed and noncondensed pairs do not mix. Details available at http: //zimp. zju. edu. cn/~qchen/Ph. DThesis/Thesis. pdf
Keep only ladder diagrams
Finite temperature:
T-matrix Formalism n T-matrix n Fermion self-energy: Q, K -- 4 -momentum
Self Energy
Self-consistent Equations n Gap equation: BEC condition n Pseudogap equation: Pair density (Boson number) n Number equation(s) Trap effect:
Summary of BEC Analogy n Pair chemical potential: determines n Total ``number” of pairs n Noncondensed pairs: …………………. Fermion pairs Ideal Point bosons
Behavior of gaps vs T How to determine Tc? --- Fraction of condensed pairs. Underlying microscopic theory for this -- PRL 81, 4708 (1998)
Critical Temperature/Trap Effects Homogeneous case In trap: use local density approx BEC BCS Spatial variation of gap, density Will excite fermions at edge, bosons in middle BEC BCS gap profile
Comparison with alternative theories • • • Good agreement on density profiles No discontinuity or nonmonotonic behavior in chemical potential (homogeneous) or density profiles (in trap) Superfluid density properly vanishes at Tc (without discontinuity or nonmonotonicity) Fermi gas at unitarity Cuprates Stajic, Chen, and Levin, PRL 94, 060401 (2005) Chen, Kosztin, Janko, and Levin, PRL 81, 4708 (1998)
Spectral function -- Effects of phase coherence Above Tc: Below Tc: Exists an zero at
What is a pseudogap? • Pseudogap becomes “real” upon phase coherence Chen, Kosztin, and Levin, PRB 63, 184519 (2001)
Population Imbalance Gap equation
Taylor Expansions
Population imbalanced superfluidity Gapless excitation spectrum ! Breached pair or Sarma state Unstable when Interaction weak Imbalance high
Three ways to accommodate polarization n Breached Pair Sarma (Sarma) State Phase Separation FFLO Phase Separation = 50 -50 SF + Normal region Rice data
Effects of population imbalance – Homogeneous case Mean-field level
Homogeneous case – Behavior of. Tc BEC Unpolarized case Phys. Rev. Lett. 97, 090402 (2006) BCS
Homogeneous phase diagram -Intermediate temperature superfluid PRL 97, 090402 (2006) Phase diagram (at T=0) I: Fermi gas; III: Sarma SF (BEC regime) IIA-B: PS+FFLO; IIC-D: Intermediate T SF, PS at low T
Part III Superfluidity From a Pseudogapped State --- Comparing theory and experiment
Cupratephase diagram QC et al, PRL 81, 4708 (1998). Apply crossover theory to d-wave lattice case BSCCO BCS : Doping Hufner et al, Rep. before Prog. Phys. 71, 062501 (2008) Pairs localize and. Tc vanishes well BEC.
Atomic Fermi gases n Equal spin mixture
Critical Temperature/Trap Effects Homogeneous case In trap: use local density approx BEC BCS Spatial variation of gap, density Will excite fermions at edge, bosons in middle BEC BCS gap profile
Thermodynamics of Fermi gases n n Bosonic contribution to thermodynamic potential Entropy: fermionic and bosonic.
Entropy of Fermi gases in a trap n n n Power law different from noninteracting Fermi or Bose gases Fall in between, power law exponent varies. Can be used to determine. T for adiabatic field sweep experiments
Phase Diagram of. Ultracold Atoms Present Theory and JILA Data Contour plot PRA 73, 041601(R) (2006) PRL 95, 260405 (2005) Theory Equilibrium phase diagram Use sweep projected temperature to plot effective. Tc or at given superfluid density.
Uncondensed Pairs Smooth out the Profiles Phys. Rev. Lett. 94, 060401 (2005) Unitary profile Data from Duke Below Tc Condensate Noncondensedpairs Fermions
Thermodynamics and pseudogap effects Duke data No fitting parameter! T* appears as temperature where 2 curves meet Science 307, 1296 (2005)
Thermodynamics and pseudogap effects n First evidence (with experiment) for a superfluid phase transition Science 307, 1296 (2005) Thermodynamic properties of strongly interacting trapped gases
Linear response theory for RF 1 -2 superfluid with 2 -> 3 transition State 3 empty
RF Spectroscopy and Pseudogap Effects C. Chin et al, Science 305, 1128 (2004). Signal strength New data at unitarity from Grimm Red line – Free atom peak
Momentum Resolved RF spectroscopy --- Counterpart of ARPES
Energy distribution curves Homogeneous “free” Fermi gas Broad peak emerges for high k as a result of Fermi function suppression Energy determined by curve fitting at high k inaccurate. QC and K. Levin, PRL 102, 190402 (2009). = 0. 62, = 0. 33, T/TF = 0. 3
Two branches – Homogeneous case Population of two branches self-consistently determined – not by hand. • Unitary, = 0. 7, pg=0. 5, = 0. 1, T/TF=0. 4 Big pseudogap and relatively high T needed.
Momentum resolved RF -- In traps T/Tc = 1. 35 1. 1 0. 8 0. 63 1. 0 0. 1
Comparison -- Energy distribution curves Fermi gas at unitarity in a trap QC and K. Levin, PRL 102, 190402 (2009). Theory, T/Tc = 1. 1 Experiment JT Stewart, JP Gaebler, DS Jin, Nature 454, 744 (2008) 0, determined by experiment
Intensity map of momentum resolved RF spectra for trapped Fermi gas at unitarity JILA data Stewart et al, Nature 454, 744 (2008) Our theory QC and K. Levin, PRL 102, 190402 (2009).
Comparison between theory and experiment -- Population imbalanced. Fermi gases
Summary of phase diagrams for Fermi gas with population imbalance n At unitarity n TOP: Strict Mean field Theory. No pairing fluctuations n BOTTOM: Effects of Pairing fluctuations. Only pairing correlations included so that phase separation stability is over-estimated. =p
Population imbalance phase diagrams PRL 98, 110404 (2007) Unitary: From profiles, MIT reports“highly correlated T>Tc normal state” In RF expts. , MIT Reports “Normal State Pairing Gap”. Polarized Superfluid N = Normal PG = pseudogap PS = Phase Separation Solid lines separate different phases.
Comparison with Rice data Unitary. Rice Phasedata Diagram Theory PRL 98, 110404 (2007)
Comparison with MIT Data at. Unitarity PRL 98, 110404 (2007). Unitary Phase Diagram Experiment Theory Trap depth roughly proportional to T, p ~ 0. 5 – 0. 6.
Conclusions n New state of superfluidity: non-condensed pairs present --pseudogap effects are evident. n Successfully applied to multiple cold atom expts and cuprates n Inclusion of effects of temperature andnoncondensedpairs is crucial to arrive at a meaningful quantitative comparison with experiments. n Lots of potential applications and interestfrom astrophysics, nuclear and even particle physics. n BCS-BEC crossover theory is thought to berelevant to high. Tc, where the pairs are small. n Where to go next: Optical lattices – beyond one trap physics
A whole new field Interface of AMO and condensed matter physics Cooper pairs of electrons in superconductors Thank you ! Excitons in semiconductors 3 He Alkali atoms in ultracold atom gases Neutron pairs, proton pairs in nuclei And neutron stars atom pairs in superfluid 3 He-A, B Mesons in neutron star matter
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